Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

3. Techniques of Differentiation

The Chain Rule

Calculus

3. Techniques of Differentiation

The Chain Rule

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2:42 minutes

Problem 3.6.3

Textbook Question

Derivative Calculations

In Exercises 1–8, given y = f(u) and u = g(x), find dy/dx = f'(g(x)) g'(x).

y = sin u, u = 3x + 1

Verified step by step guidance

First, identify the functions involved: y = sin(u) and u = 3x + 1. We need to find dy/dx using the chain rule.

The chain rule states that dy/dx = dy/du * du/dx. So, we need to find the derivatives dy/du and du/dx separately.

Calculate dy/du: Since y = sin(u), the derivative of y with respect to u is dy/du = cos(u).

Calculate du/dx: Since u = 3x + 1, the derivative of u with respect to x is du/dx = 3.

Combine the derivatives using the chain rule: dy/dx = dy/du * du/dx = cos(u) * 3. Substitute u = 3x + 1 into the expression to get dy/dx = cos(3x + 1) * 3.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Chain Rule

The chain rule is a fundamental differentiation technique used when dealing with composite functions. It states that if you have a function y = f(u) and u = g(x), then the derivative dy/dx is found by multiplying the derivative of f with respect to u, f'(u), by the derivative of u with respect to x, g'(x). This allows us to differentiate complex functions by breaking them down into simpler parts.

Derivative of Sine Function

The derivative of the sine function, sin(u), with respect to u is cos(u). This is a basic derivative rule in calculus, which is essential for solving problems involving trigonometric functions. When applying the chain rule, knowing the derivative of sin(u) helps in finding the derivative of composite functions involving sine.

Linear Function Derivative

The derivative of a linear function u = 3x + 1 with respect to x is simply the coefficient of x, which is 3. Linear functions are straightforward to differentiate, and understanding this concept is crucial when applying the chain rule to find dy/dx for composite functions where one part is linear.

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Textbook Question

b. Slopes on a tangent curve What is the smallest value the slope of the curve can ever have on the interval −2 < x < 2? Give reasons for your answer.

Textbook Question

Temperatures in Fairbanks, Alaska The graph in the accompanying figure shows the average Fahrenheit temperature in Fairbanks, Alaska, during a typical 365-day year. The equation that approximates the temperature on day x is

y = 37 sin[(2π/365)(x − 101)] + 25

and is graphed in the accompanying figure.

a. On what day is the temperature increasing the fastest?

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Textbook Question

y = 37 sin[(2π/365)(x − 101)] + 25

and is graphed in the accompanying figure.

b. About how many degrees per day is the temperature increasing when it is increasing at its fastest?

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Textbook Question

Falling meteorite The velocity of a heavy meteorite entering Earth’s atmosphere is inversely proportional to √s when it is s km from Earth’s center. Show that the meteorite’s acceleration is inversely proportional to s².

Textbook Question

Derivative Calculations

In Exercises 1–8, given y = f(u) and u = g(x), find dy/dx = f'(g(x)) g'(x).

y = cos u, u = −x/3

Textbook Question

Derivative Calculations

In Exercises 1–8, given y = f(u) and u = g(x), find dy/dx = f'(g(x)) g'(x).

y = √u, u = sin x

Textbook Question

Derivative Calculations

In Exercises 1–8, given y = f(u) and u = g(x), find dy/dx = f'(g(x)) g'(x).

y = sin u, u = x − cos x

Textbook Question

Temperature and the period of a pendulum For oscillations of small amplitude (short swings), we may safely model the relationship between the period T and the length L of a simple pendulum with the equation:

T = 2π√(L/g),

where g is the constant acceleration of gravity at the pendulum’s location. If we measure g in centimeters per second squared, we measure L in centimeters and T in seconds. If the pendulum is made of metal, its length will vary with temperature, either increasing or decreasing at a rate that is roughly proportional to L. In symbols, with u being temperature and k the proportionality constant,

dL/du = kL.

Assuming this to be the case, show that the rate at which the period changes with respect to temperature is kT/2.

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Derivative CalculationsIn Exercises 1–8, given y = f(u) and u = g(x), find dy/dx = f'(g(x)) g'(x).y = sin u, u = 3x + 1

Key Concepts

Chain Rule

Derivative of Sine Function

Linear Function Derivative

Watch next

Derivative Calculations

In Exercises 1–8, given y = f(u) and u = g(x), find dy/dx = f'(g(x)) g'(x).

y = sin u, u = 3x + 1